= Fixed-endpoint rotor fluctuation normalization
{title2=$Z_0=\sqrt{I/(2\pi\beta\hbar^2)}$}
In a temporal winding sector of a <quantum rotor>, decompose the lifted angle into a straight winding path and a fluctuation vanishing at both endpoints. The cross term in the kinetic <action> vanishes, so all sectors have the same <Gaussian path integral>. With inverse-energy imaginary time its value is the displayed diagonal free-particle kernel on the line. Integrating the initial angle separately supplies $2\pi$. Thus this fixed-endpoint factor is an amplitude density, not an additional rotor partition function or an uncounted angular zero mode.
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